Stability and Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback**Project supported by the National Natural Science Foundation of China (Grant No. 61104040), the Natural Science Foundation of Hebei Province, China (Grant No. E2012203090), and the University Innovation Team of Hebei Province Leading Talent Cultivation Project, China (Grant No. LJRC013).

Abstract

The stability and the Hopf bifurcation of a nonlinear electromechanical coupling system with time delay feedback are studied. By considering the energy in the air-gap field of the AC motor, the dynamical equation of the electromechanical coupling transmission system is deduced and a time delay feedback is introduced to control the dynamic behaviors of the system. The characteristic roots and the stable regions of time delay are determined by the direct method, and the relationship between the feedback gain and the length summation of stable regions is analyzed. Choosing the time delay as a bifurcation parameter, we find that the Hopf bifurcation occurs when the time delay passes through a critical value. A formula for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given by using the normal form method and the center manifold theorem. Numerical simulations are also performed, which confirm the analytical results.